Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63050 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,32)(4,20,16,28)(5,11,24,31)(6,12,7,13)(9,17,27,18)(10,25,26,21)(22,29,23,30)(33,34,39,36)(35,43,47,56)(37,48,50,41)(38,49,53,42)(40,52,44,46)(45,59,60,57)(51,55,63,62)(54,61,58,64);(1,3,8,19)(2,9,14,27)(4,5,16,24)(6,17,7,18)(10,11,26,31)(12,15,13,32)(20,30,28,29)(21,22,25,23)(33,35,39,47)(34,40,36,44)(37,45,50,60)(38,52,53,46)(41,54,48,58)(42,56,49,43)(51,64,63,61)(55,59,62,57);(1,4,8,16)(2,10,14,26)(3,5,19,24)(6,21,7,25)(9,11,27,31)(12,28,13,20)(15,29,32,30)(17,22,18,23)(33,36,39,34)(35,44,47,40)(37,48,50,41)(38,42,53,49)(43,52,56,46)(45,58,60,54)(51,55,63,62)(57,64,59,61);(1,5)(2,11)(3,16)(4,19)(6,23)(7,22)(8,24)(9,26)(10,27)(12,30)(13,29)(14,31)(15,28)(17,21)(18,25)(20,32)(33,37)(34,41)(35,45)(36,48)(38,51)(39,50)(40,54)(42,55)(43,57)(44,58)(46,61)(47,60)(49,62)(52,64)(53,63)(56,59);(1,6,8,7)(2,12,14,13)(3,17,19,18)(4,21,16,25)(5,22,24,23)(9,15,27,32)(10,28,26,20)(11,29,31,30)(37,50)(41,48)(45,60)(51,63)(54,58)(55,62)(57,59)(61,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259216);